Reduction in the Thermal Conductivity ofSingle Crystalline Silicon by Phononic CrystalPatterningPatrick E. Hopkins,† Charles M. Reinke,† Mehmet F. Su,‡ Roy H. Olsson III,†Eric A. Shaner,† Zayd C. Leseman,‡ Justin R. Serrano,† Leslie M. Phinney,† andIhab El-Kady*,†,‡

†Sandia National Laboratories, Albuquerque, New Mexico 87185, United States, and ‡University of New Mexico,Albuquerque, New Mexico 87131, United States

ABSTRACT Phononic crystals (PnCs) are the acoustic wave equivalent of photonic crystals, where a periodic array of scatteringinclusions located in a homogeneous host material causes certain frequencies to be completely reflected by the structure. In conjunctionwith creating a phononic band gap, anomalous dispersion accompanied by a large reduction in phonon group velocities can lead toa massive reduction in silicon thermal conductivity. We measured the cross plane thermal conductivity of a series of single crystallinesilicon PnCs using time domain thermoreflectance. The measured values are over an order of magnitude lower than those obtainedfor bulk Si (from 148 W m-1 K-1 to as low as 6.8 W m-1 K-1). The measured thermal conductivity is much smaller than that predictedby only accounting for boundary scattering at the interfaces of the PnC lattice, indicating that coherent phononic effects are causingan additional reduction to the cross plane thermal conductivity.

KEYWORDS Phononic crystal, thermal conductivity, phonon transport, incoherent vs. coherent effects

Silicon is at the heart of almost all hi-tech devices andapplications. It is arguably the seed of the semicon-ductor revolution. One of the fundamental tenets for

silicon electronics is controlling the heat flow that is abyproduct of the nature of electronic operations. Whileultralow thermal conductivity has been observed in siliconnanowires and nanomeshes, the fundamental barriers re-main against the practical implementation of such ultrasmalldevices. Here, we report on a successful methodologyimplementing a phononic crystal geometry in silicon thatresults in the same order of magnitude of thermal conduc-tivity reduction as silicon nanowires while maintaining thecharacteristic length scales at an order of magnitude larger.Since phononic crystals can be mass produced and arecompatible with standard CMOS fabrication, this enables thepractical implementation of such devices. An added bonusof this approach comes to light by realizing that the electronmean free path is an order of magnitude smaller than thatof the phonons involved, thereby possibly laying the founda-tion for the realization of exceptionally high ZT (thermoelec-tric figure of merit) in silicon and other phononic crystal(PnC) amenable material systems.

In general, material systems with structural length scaleson the order of nanometers have unique abilities to controlthermal transport.1 Internal interfaces and boundaries innanosystems create thermal carrier scattering events, and

tailoring the period or structure of these boundaries offers aunique method for tuning their thermal properties. Whilethis aspect of reducing the thermal conductivity alone hasresulted in ultralow thermal conductivity of fully densematerials2,3 which proves useful for thermal barrier ap-plications,4 this “boundary engineering” has also provensuccessful in designing nanocomposites for thermoelec-tric applications.5 The efficiency of material systems forthermoelectric applications can be quantified with thewell-known nondimensional thermoelectric figure of merit,ZT ) S2σTκ-1, where T is the temperature, S is the Seebeckcoefficient, σ is the electrical conductivity, and κ is thethermal conductivity. Therefore, introducing interfacesand boundaries at length scales that will scatter phononsmore frequently than electrons will reduce the thermalconductivity more than the electrical conductivity, therebyincreasing ZT.

This approach of nanoengineering material boundarieshas been useful in decreasing the thermal conductivity ofsingle crystalline silicon. Silicon nanowires have shownparticular promise for low thermal conductivity applicationsthrough further reduction of nanowire diameter and in-creased surface roughness.6-9 However, structural stabilityand large contact areas are necessary criteria for mostapplications, and individual nanowires lack both of thesecharacteristics. Increasing the perpendicular contact area ofindividual nanowires causes single nanowires to lose theirunique thermal properties, and creating large arrays ofnanowires with appropriate lengths creates difficulties innanowire alignment and integrity. Therefore, we propose a

* Corresponding author, ielkady@sandia.gov.Received for review: 8/17/2010Published on Web: 11/24/2010

pubs.acs.org/NanoLett

© 2011 American Chemical Society 107 DOI: 10.1021/nl102918q | Nano Lett. 2011, 11, 107–112

new parameter to evaluate the applicability of low thermalconductivity materials based on the usable area of thematerial of interest. This parameter, the thermal conductiv-ity of a “unit cell” of a material, is given by κuc ) κm/Auc,where κm is the measured thermal conductivity of thenanosystems and Auc is the minimum, repeatable crosssectional area of the solid matrix of the unit cell. Ideally, Auc

should be as large as possible so that κuc is minimized forlow thermal conductivity applications. In the previouslymeasured nanowires, Auc is defined as the cross section areaof the nanowire.

Following this logic, recent work by Yu et al.10 studiedthe thermal conductivity of Si nanomesh films that weretheorized to have phononic crystal properties. Phononiccrystals are the acoustic wave equivalent of photonic crys-tals, where a periodic array of scattering inclusions locatedin a homogeneous host material causes certain frequenciesto be reflected by the structure (for a review, see ref 11). Thenanomeshes studied by Yu et al. exhibited thermal conduc-tivities similar to those of the lowest thermal conductivitynanowires. These nanomeshes were able to be developedas films with a 100 µm2 areal footprint, alleviating theaforementioned stability aspect of the nanowires, but thenanomesh films were only grown to ∼20 nm film thick-nesses. Therefore, the unit cell area as previously defined isstill of the same order of magnitude as many of the Sinanowires. In the case of the nanomeshes, Auc is defined asin-plane, cross sectional area of an individual ligament in thenanomesh, explicitly defined by Yu et al.10

In response to the previously mentioned research, in thiswork, we investigate the thermal conductivity of singlecrystalline silicon PnCs with a thickness of 500 nm, porespacings of several hundreds of nanometers, and arealfootprints of ∼10000 µm2. We measure the thermal con-ductivity in the cross plane direction of these PnCs with timedomain thermoreflectance; this direction of thermal propa-gation exhibits Auc nearly 3 orders of magnitude larger thanthat of the nanomeshes. Although the PnCs studied in thiswork have band gaps in the gigahertz regime, well belowterahertz phonon frequencies known to affect thermaltransport in silicon,12 the periodic nature of the PnCs coher-ently alter the phononic spectrum, which affects the thermalconductivity. In addition, incoherent phonon scattering atthe physical boundaries of the PnC lattice will also cause areduction in the phonon thermal conductivity. We use thisto describe the thermal conductivity reduction in the PnCsstudied in this work by accounting for phonon scattering anddispersion changes in the specific PnCs examined in thisstudy.

The fabrication of the PnC begins with 150 µm silicon-on-insulator (SOI) wafers, where the buried oxide (BOX)layer is 3 µm thick. The ⟨100⟩, n-type, top Si layer where thePnC devices are realized is 500 nm thick and has a resistivityof 37.5-62.5 Ω·cm. The PnCs are formed by etchingcircular air holes of diameter d ) 300-400 nm in the top

Si, arranged in a simple cubic lattice, with center-to-centerhole spacings, a, of 500, 600, 700, and 800 nm. The samplesstudied here specifically have: d/a ) 300/500 nm (3/5), 300/600 nm (3/6), 400/700 nm (4/7), and 400/800 nm (4/8). Inthe cross plane direction, these PnCs have solid matrix unitcell areas of Auc ) a2 - πd2/4. Release areas are also etchedin the top Si to the BOX, and the PnC membranes aresuspended above the substrate by removing the BOX invapor phase hydrofluoric acid (VHF). Figure 1a shows a top-down image of a membrane containing two PnCs with a )500 nm and d ) 300 nm. The membrane is 60 µm wideand 200 µm long and is comprised of two PnCs separatedby a 20.5 µm wide unpatterned area. The length of each PnCis 80 µm for a total of 160 periods when a ) 500 nm. The600, 700, and 800 nm lattice constant devices have thesame membrane width, length, and spacing between thePnCs and contain 133, 115, and 101 PnC periods, respec-tively, maintaining a nearly constant PnC length of 80 µmfor each sample. Figure 1b shows a side image of a PnCmembrane and its suspension above the substrate to isolatethermal effects in the membrane. Figure 1c shows a closein image of a Si/air PnC.

We measured the thermal conductivity of the PnCs withthe time-domain thermoreflectance technique (TDTR).13,14

Our specific experimental setup is described in detail in ref15. TDTR is a noncontact, pump-probe technique in whicha modulated train of short laser pulses (in our case ∼100 fs)is used to create a heating event (“pump”) on the surface ofa sample. This pump-heating event is then monitored witha time-delayed probe pulse. The change in the reflectivityof the probe pulses at the modulation frequency of the pumptrain is detected through a lock-in amplifier; this change in

FIGURE 1. (a) Top-down image of a membrane containing two PnCswith a ) 500 nm and d ) 300 nm. The membrane is 60 µm wideand 200 µm long. A 20.5 µm wide area between the PnCs is locatedin the center of the membrane. The length of each PnC is 80 µm fora total of 160 periods. (b) Side image of a PnC membrane showingits suspension above the substrate. (c) Close in image of a Si/air PnCshowing the lattice constant, a, and hole diameter, d. In the crossplane direction, these PnCs have solid matrix unit cell areas of Auc

) a2 - πd2/4.

© 2011 American Chemical Society 108 DOI: 10.1021/nl102918q | Nano Lett. 2011, 11, 107-–112

reflectivity is related to the temperature change on thesurface of the sample. These temporal temperature data arerelated to the thermophysical properties of the sample ofinterest. In practice, a thin metal film is deposited on thesample of interest which acts as a thermometer that absorbsthe pump energy in less than 15 nm below the surface. Inthis study, we sputter 100 nm of Al on the surface of thePnCs. We monitor the thermoreflectance signal over 4.0 nsof probe delay time. The deposited energy takes ∼100 psto propagate through the Al layer, and the remaining 3.9 nsof delay time are related to the heat flow across the Al/PnCinterface and through the PnC.

The thermoreflectance signal we monitor is the ratio ofthe in-phase to the out-of-phase voltage recorded by the lock-in amplifier. The ratio is related to the temperature changeby

where ∆T is the temperature rise on the sample surface, ω0

is the pump modulation frequency, ωL is the modulationfrequency of the laser source, and τ is the pump-probedelay time. Our Ti:sapphire oscillator has a nominal repeti-tion rate of 80 MHz. The thermal model and analysis usedto predict ∆T is described in detail in refs 16 and 17. In short,the thermal model accounts for heat transfer in compositeslabs18 from a periodic, Gaussian source (pump) convolutedwith a Gaussian sampling spot (probe).16 In our experiments,our pump modulation frequency is 11 MHz and our pumpand probe spot sizes are 15 µm. The temperature change atthe surface is related to the thermal conductivity, κ, and heatcapacity, C, of the composite slabs and the thermal boundaryconductance, hK, between each slab at a distance of

δ ) √2κ/(Cω0)

underneath the surface, where δ is the thermal penetrationdepth from the modulated pump train.

As hK is highly dependent on the structure and materialcomposition around the interface, for any given materialsystem hK can change from sample to sample.19-21 There-fore, we deposit 100 nm of Al on the entire wafer includingboth the PnC structures and the areas without the PnCs. Thisallows us to independently measure hK with TDTR at thedeposited Al/Si interface without any complication from thePnC structure. (See Supporting Information for measure-ments and sensitivity analysis.) We measure hK at the Al/Simaterial interfaces as 170 ( 20 MW m-2 K-1. We thencollect TDTR data on the four different PnC structures (3/5,

3/6, 4/7, and 4/8) at room temperature. The temporal decayof the thermoreflectance signal (which is related to thetemperature change) is much different than that of the Sifilm used for hK calibration (see Supporting Information). Amore shallow decay in the thermoreflectance signal repre-sents a longer thermal time constant, which implies areduction in the thermal diffusivity. This qualitatively showsthe change in the thermal diffusivity of the PnC structurescompared to the unpatterned Si thin film. To quantify this,we account for the reduction in C and hK due to surfaceporosity of the Al-coated PnC structure by multiplying thebulk values of C in the Al film and Si PnC and hK at the Al/SiPnC interface by a factor of (1 - φ) where φ is the porosity;this effectively treats the air in the PnC as nonabsorbing sothat our best fit thermal conductivities represent the thermaltransport through only the solid matrix of the PnC. We takethe bulk values of C from the literature22 and take the“nonporous” hK value from the previous measurement atthe Al/Si interface in the non-PnC fabrication locations onthe wafer. The porosities of the 3/5, 3/6, 4/7, and 4/8structures are calculated based on the geometry of eachstructure, and are φ ) 0.28, 0.19, 0.25, and 0.19, respec-tively. The measured thermal conductivities on the (3/5),(3/6), (4/7), and (4/8) PnC structures are 5.84 ( 1.3, 4.81 (1.0, 7.11 ( 1.8, and 6.58 ( 0.5 W m-1 K-1, where theuncertainty represents the standard deviation among thebest fit to three different data sets taken on three differentsamples with similar geometries (nine data sets total).

Figure 2 shows the measured thermal conductivity of thefour PnCs as a function of unit cell area along with the

ratio ) -

Re[ ∑M)-∞

∞

∆T(ω0 + MωL) exp[iMωLτ]]

Im[ ∑M)-∞

∞

∆T(ω0 + MωL) exp[iMωLτ]]

(1)

FIGURE 2. Measured thermal conductivity as a function of unit cellsolid area. The four PnCs are depicted by the unfilled squares andthe Si nanowires by the downward triangles,9 upward triangles,7 andcircles.6 The recent nanowire arrays and nanomesh data are de-picted by the diamonds.10 All data shown in this figure are for roomtemperature measurements except for the nanomesh data whichwas taken at 280 K. The solid lines represent K ) Kuc_PnC_avgAuc )Km_PnC_avgAuc/Auc_PnC_avg, where we calculate Km_PnC_avg and Auc_PnC_avg

by averaging values from the PnCs (Km_PnC_avg ) 6.09 W m-1 K-1 andAuc_PnC_avg ) 0.337 µm-2). The resulting Kuc on the PnCs is the lowestKuc of any nanostructured Si material.

© 2011 American Chemical Society 109 DOI: 10.1021/nl102918q | Nano Lett. 2011, 11, 107-–112

measured thermal conductivity of the various Si nanostruc-tures from previous studies. For the unit cell area of thePnCs, we subtract the area of the air holes to include onlythe unit cell of the solid fraction of the PnC. The thermalconductivities of the PnC structures show a drastic reductionfrom that of bulk, single crystalline Si (148 W m-1 K-1).22

Although the nanowires and nanomesh data show similar,if not greater reduction, the PnCs have a much greater (1-3orders of magnitude) unit cell solid area than the nanowiresfor the same thermal conductivity. To directly compare thePnCs to the nanowire data, consider κuc on the PnCs ex-tended to lower dimensions. The solid line in Figure 2represents

κ ) κuc_PnC_avgAuc/Auc_PnC_avg

where we calculate κm_PnC_avg and Auc_PnC_avg by averagingvalues from the PnCs. This effectively projects the “deviceapplicability” of the PnCs to lower unit cell areas. As appar-ent from Figure 2, κuc on the PnCs is the lowest κuc of anysilicon nanostructured material (Auc < 10-12 m).

To understand the origin of this thermal conductivityreduction at room temperature, we turn to the Callaway-Holland-type model,23,24 given by

where Cj is the specific heat per normal mode at frequencyω(q), vj is the phonon velocity, τj is the scattering time, andq is the wavevector. In bulk Si, the scattering time aroundroom temperature is dominated by Umklapp processes, witha relatively small contribution from impurity scattering.From Matthiessen’s rule, the scattering time is given by τj(q)) (τU,j

-1(q) + τI,j-1(q))-1, where the Umklapp scattering rate

is given by τU,j-1(q) ) ATω2(q) exp[-B/T], where T is the

temperature and A and B are coefficients to be determined,and the impurity scattering rate is given by τI,j

-1(q) ) Dω4(q),where D ) 1.32 × 10-45 s3 (ref 24). We fit eq 2 to themeasured thermal conductivity of bulk Si (ref 25), iteratingA and B in the Umklapp scattering rate to achieve a best fit.From this, we determine that A ) 1.4 × 10-19 s K-1 and B) 152 K. For these calculations, we use the dispersion of bulkSi by fitting a fourth degree polynomial to the dispersioncalculated by Weber.26 After determining the Umklapp scat-tering rate in Si, we then introduce a boundary scattering termvia Matthiessen’s rule to account for phonon scattering at thePnC pore boundaries given by τB,j

-1(q) ) L/vj(q), where L is theaverage distance between pore boundaries; note that thisapproach for modeling the reduction in thermal conductivitydue to boundary scattering has proven successful in predictingthe thermal conductivity of microporous,27 polycrystalline,28,29

and nanowire30 silicon samples. For the PnCs of interest in thisstudy, the pore edge to pore edge distances are 200, 300, 300,and 400 nm for the (3/5), (3/6), (4/7), and (4/8) structures,respectively.

The thermal conductivity at room temperature as afunction of L calculated via eq 2 is shown in Figure 3.Ultimately, we are interested in the thermal conductivity ofthe solid matrix in the PnC, not the reduction due to theremoval of the material to create the PnC. To directlycompare the thermal conductivity reduction of the solidmaterial in the PnC to the reduction predicted from bound-ary scattering via eq 2, we use the expression derived byEucken for the thermal conductivity of cylindrical poroussolids.31 Note that this expression has been used successfullyto account for the reduction in thermal conductivity inmicroporous solids.27 Following Eucken, the predicted ther-mal conductivity of the PnCs using eq 2 is related to thethermal conductivity of the solid matrix in the PnC through

κs ) κm(1 + 2φ/3)/(1 - φ)

Although more rigorous treatments of porosity have beenderived for nanoporous solids,32,33 due to the large poreseparation in our PnCs and relatively large porosities, weexpect that the majority of the phonon modes will bescattered diffusively at the pore boundaries (i.e., the ballisticcharacter of phonon transport and its interaction with thepore edges is not important)34 and therefore the classicalEucken treatment should hold.

The thermal conductivities of the PnCs shown in Figure3 as a function of pore-edge separation are the values from

κ ) 1

6π2 ∑j∫q

Cj(q)vj2(q)τj(q) dq (2)

FIGURE 3. The thermal conductivity of Si structures at room tem-perature as a function of L for the PnCs (unfilled squares), mi-croporous solids (filled pentagons),35 nanomesh (filled diamond),10

and a suspended 500 nm thick Si filmsthat is, an unpatterned Sislab (unfilled circle). The measured thermal conductivities of theporous structures are multiplied by a factor of [(1 + 2O/3)/(1 - O)]to account for the porosity of the structures, and thereby directlycompare the thermal conductivity of the solid matrix in the porousstructures to the model in eq 2. The solid line represents predictionsof eq 2 at room temperature as a function of L. Equation 2 predictsthe thermal conductivity of the microporous solids well and onlyslightly overpredicts the thermal conductivity of the 500 nm sus-pended film. This model, however, drastically overpredicts the PnCmeasurements. The dashed line represents predictions of the PnCthermal conductivity, based on eq 2 with DOS calculations usingthe PWE method.

© 2011 American Chemical Society 110 DOI: 10.1021/nl102918q | Nano Lett. 2011, 11, 107-–112

Figure 2 multiplied by the Eucken factor, where the porosi-ties of the PnCs are 0.28, 0.20, 0.26, and 0.20 for the (3/5),(3/6), (4/7), and (4/8) structures, respectively. For compari-son, we also include the measured thermal conductivity ofthe Si microporous solids measured by Song and Chen35 andthe nanomesh sample by Yu et al.,10 also multiplied by theEucken factor. Equation 2 predicts the thermal conductivityof the microporous solids well; however, this model over-predicts the measured PnC data by a factor of 5-7 and thedata by Yu et al. by a factor of 4. The order of magnitudereduction in the PnC thermal conductivity compared to eq2 can be ascribed to the PnC periodically porous structurechanging the Si mode density. This shift in mode density canlead to a reduction in thermal conductivity. To verify this,we measure the thermal conductivity of suspended, unpat-terned Si slabs (i.e., suspended Si slabs described earlierwithout PnC patterning). The slabs, which are 500 nm thick,are measured and analyzed via the TDTR procedure de-scribed earlier. The thermal conductivity of the 500 nmsuspended Si film is 39.2 ( 4.8 W m-1 K-1, as shown inFigure 3; the limiting dimension of this suspended film is500 nm since the suspended film boundary forces phononscattering at a distance of 500 nm below the surface. Thethermal conductivity of the suspended, upatterned film,which is only slightly overpredicted by eq 2, is nearly a factorof 3-4 higher than the PnC data which has a similar limitingspacing. This further shows that the coherent, phononiceffect in the PnCs is causing an additional reduction in thephononic thermal conductivity beyond the effect fromboundary scattering alone.

To investigate this effect further, we implemented theplane-wave expansion (PWE) technique36 to calculate thephononic density of states (DOS) of the PnC from its disper-sion and used these data to calculate the change in thermalconductivity of the PnC as compared to bulk (i.e., unpat-terened) Si. The PWE model is solved for the eigenmodesof an infinite 2D PnC structure. Such a model accuratelymatches the behavior of the experimental measurements,since the thermal waves excited by the modulated pumppropagate only a very short depth into the PnC slab due tothe high repetition rate (11 MHz) used in the TDTR experi-ments and thus do not “see” the finite thickness of thesample. The DOS of a given structure is calculated bynumerically integrating the number of modes with respectto frequency for all directions in the first 2D Brillouin zone.Figure 4 shows the integrated density of states as a functionof frequency for bulk Si and a PnC with d/a ) 0.6 (the bulkdispersion was calculated using the same material param-eters as the PnC but with d/a ) 0). The inset shows thecalculated density of states of the PnC. The observed spikein the low frequency modes is indicative of a large reductionin the phonon group velocity in the PnC lattice.

Once the DOS was found, the thermal conductivity wascalculated via eq 2. The thermal conductivity of the PnCpredicted from the PWE DOS versus L is shown in Figure 3

as a dashed line. The curve shows excellent agreement withthe measured values from the PnC samples, particularly the3/5 sample, which has the same pore radius as the valueused in the simulations. The predictions also show excellentagreement with the thermal conductivity of Yu et al.’snanomesh sample,10 indicating that the further reductionobserved in the PnC and nanomesh beyond that of eq 2using a bulk dispersion is due to the changes in the modedensity. This trend is based on calculations at smaller valuesof L, since the PWE technique becomes too computationallyintensive for larger values of L to calculate the modaldispersion up to meaningful values using the resourcesavailable at the time of publication. This is directly relatedto the fact that thermal energy in Si follows a Bose-Einsteindistribution, and thus the majority of the energy is carriedby phonons in the 1-6 THz range.12,27 Since the size of thePWE computational space grows nonlinearly with maximummode frequency, this severely limits the largest latticeconstant that can be simulated, as the frequency scalesdirectly with the PnC lattice constant. However, the soundagreement between the predicted reduction in κ of the PnCsand that measured with TDTR elucidates the role of phonondispersion and mode density on the further reduction inthermal conductivity of PnCs beyond that considering onlyboundary scattering effects.

In summary, we have experimentally studied the thermaltransport processes in single crystalline silicon phononiccrystals with submicrometer geometries. The measuredvalues are over an order of magnitude lower than that of bulkSi, and for the structures in this work represent the loweston thermal conductivity to solid area ratio of any siliconnanostructured material measured to date. The magnitude

FIGURE 4. Integrated density of states as a function of frequencyfor bulk Si (black) and PnC lattice with d/a ) 0.6 (red). The bulkdispersion was calculated using the same material parameters asthe PnC but with d/a ) 0. Inset shows the calculated density of statesof the PnC. The observed spike in the low frequency modes isindicative of a large reduction in the phonon group velocity in thePnC lattice.

© 2011 American Chemical Society 111 DOI: 10.1021/nl102918q | Nano Lett. 2011, 11, 107-–112

of this measured thermal conductivity reduction is muchlarger than that predicted from accounting for phonon-boundary scattering at the interfaces of the PnC lattice alone.To investigate the origin of this further reduction, we imple-mented the plane-wave expansion technique to calculate thephononic density of states of the PnC. The PnC density ofstates is drastically altered compared to bulk, and thepredictions of the PnC thermal conductivity agree well withthe experimental measurements, indicating that the thermalconductivity is drastically affected by the altered dispersionintroduced by the periodically nanostructed nature of phonon-ic crystals.

Acknowledgment. The authors are grateful for fundingfrom the LDRD program office. P.E.H. is grateful for fundingthrough the Sandia National Laboratories Harry S. TrumanFellowship. Sandia National Laboratories is a multiprogramlaboratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation,for the U.S. Department of Energy’s National Nuclear Secu-rity Administration under Contract DE-AC04-94AL85000.

Supporting Information Available. Additional informa-tion on phononic crystal fabrication, TDTR data analysis, andthermal conductivity modeling. This material is available freeof charge via the Internet at http://pubs.acs.org.

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